CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A ring is reduced exactly when zero is an intersection of primes

Statement

Assume the Axiom of Choice.

A commutative ring $R$ is reduced if and only if its zero ideal is the intersection of its prime ideals.

Facts & Assumptions

Given: A commutative ring $R$ and the Axiom of Choice.

[L1]

$R$ is reduced exactly when $Nil (R) = (0)$ (The nilradical and reduced rings).

[L2]

The nilradical is the intersection of all prime ideals (The nilradical is the intersection of all prime ideals).

Proof

technique · direct

1.1L1L2

If $R$ is reduced, then $Nil (R) = (0)$ by [L1]. Applying [L2] yields $(0) = ⋂_{p \in Spec R} p$ .

1.2L1L2

Conversely, if $(0) = ⋂_{p \in Spec R} p$ , then [L2] shows that $Nil (R) = (0)$ . Now [L1] gives that $R$ is reduced.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the equivalence.

Depends on

Used by

Smooth orbits are locally closed and their orbit maps are faithfully flat over every field Lemma

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §2 Ideals (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §2 Ideals (standard reference, not scraped)